Water Jug Problem using Memoization
Given two jugs with the maximum capacity of m and n liters respectively. The jugs don’t have markings on them which can help us to measure smaller quantities. The task is to measure d liters of water using these two jugs. Hence our goal is to reach from initial state (m, n) to final state (0, d) or (d, 0).
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Input: 4 3 2
Output: (0, 0) –> (4, 0) –> (4, 3) –> (0, 3) –> (3, 0) –> (3, 3) –> (4, 2) –> (0, 2)
Input: 5 2 4
Output: (0, 0) –> (5, 0) –> (5, 2) –> (0, 2) –> (2, 0) –> (2, 2) –> (4, 0)
Approach: An approach using BFS has been discussed in the previous post. In this post an approach using memoization and recursion has been discussed. At any point, there can be a total of six possibilities:
- Empty the first jug completely
- Empty the second jug completely
- Fill the first jug
- Fill the second jug
- Fill the water from the second jug into the first jug until the first jug is full or the second jug has no water left
- Fill the water from the first jug into the second jug until the second jug is full or the first jug has no water left
Approach: Using Recursion, visit all the six possible moves one by one until one of them returns True. Since there can be repetitions of same recursive calls, hence every return value is stored using memoization to avoid calling the recursive function again and returning the stored value.
Below is the implementation of the above approach:
Steps: 0 0 4 0 4 3 0 3 3 0 3 3 4 2 0 2
Time complexity: O(M * N)
Auxiliary Space: O(M * N)